Modules over trusses vs modules over rings: direct sums and free modules

TL;DR

This paper explores the categorical constructions of modules over trusses, comparing them with modules over rings, and investigates properties of free modules, direct sums, and extensions to ring-like structures.

Contribution

It provides explicit descriptions of free heaps, direct sums, and free modules over trusses, and analyzes their properties and relationships with modules over rings.

Findings

Direct sum of non-empty Abelian heaps is always infinite.

Only free rank-one modules are free over the associated truss.

Finitely generated free modules over a ring-associated truss are also free as modules over the ring.

Abstract

Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and $\mathbb{Z}$. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free modules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring.